9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 1/9 Jim Stiles The Univ. of Kansas Dept. of EECS 2-5 The Calculus of Scalar and Vector Fields (pp.33-55) Fields are functions of coordinate variables (e.g., x, ρ, θ) Q: How can we integrate or differentiate vector fields ?? A: There are many ways, we will study: 1. 4. 2. 5. 3. 6. A. The Integration of Scalar and Vector Fields 1. The Line Integral9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 2/9 Jim Stiles The Univ. of Kansas Dept. of EECS ()Ccrd⋅∫A A Q1: HO: Differential Displacement Vectors A1: HO: The Differential Displacement Vectors for Coordinate Systems Q2: A2: HO: The Line Ingtegral Q3: HO: The Contour C A3: HO: Line Integrals with Complex Contours Q4:9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 3/9 Jim Stiles The Univ. of Kansas Dept. of EECS HO: Steps for Analyzing Line Integrals A4: Example: The Line Integral 2. The Surface Integral Another important integration is the surface integral: ()sSrds⋅∫∫A Q1: A1: HO: Differential Surface Vectors HO: The Differential Surface Vectors for Coordinate Systems Q2: A2: HO: The Surface Integral9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 4/9 Jim Stiles The Univ. of Kansas Dept. of EECS Q3: HO: The Surface S A3: HO: Integrals with Complex Surfaces Q4: HO: Steps for Analyzing Surface Integrals A4: Example: The Surface Integral 3. The Volume Integral The third important integration is the volume integral—it’s the easiest of the 3! ()Vgrdv∫∫∫ Q1:9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 5/9 Jim Stiles The Univ. of Kansas Dept. of EECS A1: HO: The Differential Volume Element HO: The Volume V Example: The Volume Integral B. The Differentiation of Vector Fields 1. The Gradient The Gradient of a scalar field ()gr is expressed as: ()gr∇ ()()grr∇= A Q: A: HO: The Gradient9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 6/9 Jim Stiles The Univ. of Kansas Dept. of EECS Q: A: HO: The Gradient Operator in Coordinate Systems Q: The gradient of every scalar field is a vector field—does this mean every vector field is the gradient of some scalar field? A: HO: The Conservative Field Example: Integrating the Conservative Field 2. Divergence The Divergence of a vector field ()rA is denoted as: ()r∇⋅A ()()rgr∇⋅ =A9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 7/9 Jim Stiles The Univ. of Kansas Dept. of EECS Q: A: HO: The Divergence of a Vector Field Q: A: HO: The Divergence Operator in Coordinate Systems HO: The Divergence Theorem 3. Curl The Curl of a vector field ()rA is denoted as: ()r∇×A ()()rr∇× =AB9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 8/9 Jim Stiles The Univ. of Kansas Dept. of EECS Q: A: HO: The Curl Q: A: HO: The Curl Operator in Coordinate Systems HO: Stoke’s Theorem HO: The Curl of a Conservative Vector Field 4. The Laplacian ()2gr∇9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 9/9 Jim Stiles The Univ. of Kansas Dept. of EECS C. Helmholtz’s Theorems ()()and/orrr∇⋅ ∇×AA Q: A: HO: Helmholtz’s Theorems9/6/2005 Differential Line Vector Elements.doc 1/4 Jim Stiles The Univ. of Kansas Dept. of EECS Differential Displacement Vectors The derivative of a position vector r, with respect to coordinate value A (where {},,,,,,xyz rρφθ∈A ) is expressed as: ()()()()ˆˆˆˆˆˆˆˆˆxyzyxzxyzdr dxyzdddydx dzddddydx dzddd=++=++⎛⎞⎛⎞ ⎛⎞=++⎜⎟ ⎜⎟⎜⎟⎝⎠ ⎝⎠⎝⎠AAAAAAAAaaaaaaaaa A: The vector above describes the change in position vector r due to a change in coordinate variable A . This change in position vector is itself a vector, with both a magnitude and direction. Q: Immediately tell me what this incomprehensible result means or I shall be forced to pummel you !9/6/2005 Differential Line Vector Elements.doc 2/4 Jim Stiles The Univ. of Kansas Dept. of EECS For example, if a point moves such that its coordinate A changes from A to +∆AA, then the position vector that describes that point changes from r to + r∆A . In other words, this small vector ∆A is simply a directed distance between the point at coordinate A and its new location at coordinate +∆AA! This directed distance ∆A is related to the position vector derivative as: ˆˆˆ xyzdrddydx dzaaaddd∆=∆⎛⎞⎛⎞ ⎛⎞=∆ +∆ +∆⎜⎟ ⎜⎟⎜⎟⎝⎠ ⎝⎠⎝⎠AAAAAAAAA As an example, consider the case when ρ=A . Since cosxρφ= and sinyρφ= we find that: r + r∆A∆A9/6/2005 Differential Line Vector Elements.doc 3/4 Jim Stiles The Univ. of Kansas Dept. of EECS ()()ˆˆˆˆˆˆˆˆˆcos sincos sinxyzxyzxydydr dx dzaaadd d ddddzaaadddaaaρρρ ρ ρρφ ρφρρρφφ=++=++=+= A change in position from coordinates ,,zρφ to ,,zρρφ+∆ results in a change in the position vector from r to + r∆A . The vector ∆A is a directed distance extending from point ,,zρφ to point ,,zρρφ+∆ , and is equal to: ˆˆˆrdcos sinxydaaaρρρρφ ρφρ∆=∆=∆ +∆=∆A If ∆A is really small (i.e., as it approaches zero) we can define something called a differential displacement vector dA : rˆpaρ∆=∆A y x9/6/2005 Differential Line Vector Elements.doc 4/4 Jim Stiles The Univ. of Kansas Dept. of EECS 00limlimddrddrdd∆→∆→∆⎛⎞=∆⎜⎟⎝⎠⎛⎞=⎜⎟⎝⎠AAA AAAAA For example: ˆrdddaddρρρρρ== Essentially, the differential line vector dA is the tiny directed distance formed when a point changes its location by some tiny amount, resulting in a change of one coordinate value A by an equally tiny (i.e., differential) amount dA . The directed distance between the original location (at coordinate value A ) and its new location (at coordinate value d+AA) is the differential displacement vector dA . We will use the differential line vector when evaluating a line integral. r + rdA dA9/6/2005 The Differential Line Vector for Coordinate Systems.doc 1/3 Jim Stiles The Univ. of Kansas Dept. of EECS The Differential Displacement Vector for Coordinate Systems Let’s determine the differential displacement vectors for each coordinate of the Cartesian, cylindrical and spherical coordinate systems! Cartesian This